K. PANAGIOTIDOU ET AL.
Owing to (29), we consider , (W) the open
subset of points , where in a neighbor-
hood of every Q.
Lemma 3.3 Let M be a real hypersurface in
equipped with Jacobi operator satisfying (1). Then
Proof: Due to (29) we obtain: on 1
Differentiation of the last relation along
into account (19), (20) and yields: ,
which is a contradiction. Therefore, is empty. □
On W, because of Lemma 3.3, we have , hence
the relations (17), (19) and (20) become:
Using the last relations and Lemma 3.2 we obtain:
Combining the last two relations we have:
Let 2, (2) be the set of points W
which there exists a neighborhood of every Q such that
. So from (30) we have: 16 .
Differentiating the last relation with respect to
taking into account (21), (22), (27) and (28), we have:
, which is impossible. So 2 is empty. Hence,
on W we have . Then, relations (21) and
(28), because of (27) imply respectively:
. Relation (26), because of
and (27) yields: 1
. Substitution of 1
. Differentiation of
the last one along U
and taking into account (22) leads
, which is a contradiction. So we obtain the
Proposition 3.4 Let M be a real hypersurface in
c, equipped with Jacobi operator satisfying (1).
Then M is a Hopf hypersurface.
4. Proof of Main Theorem
Since M is a Hopf hypersurface, we can write
, being a local or-
thonormal basis and the following relation holds:
(Corollary 2.3 ). Furthermore,
due to Theorem 2.1 , we have that
From (5), due to
The relation (1), because of (31) implies:
and , for
. Combining the last two relations leads to:
, M is locally
, in case of
congruent to a tube of radius π
over a holomorphic
due to  and to a Hopf hypersurface
, the last relation im-
and we obtain:
which because of Theorem A completes the proof of
The first author is granted by Foundation Alexandros S.
Onasis. Grant Nr: G ZF 044/2009-2010. The authors
would like to thank Professors Juan de Dios Perez and
Patrick J. Ryan. Their answers improved the proof of
main theorem. Finally, the authors express their grati-
tude to the referee who has contributed to improve the
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